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    <title>dt_ility</title>
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    <center>Scilab Function</center>
    <div align="right">Last update : April 1993</div>
    <p>
      <b>dt_ility</b> -  detectability test</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>[k, [n [,U [,Sld ] ] ]]=dt_ility(Sl [,tol])  </tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Parameters</font>
    </h3>
    <ul>
      <li>
        <tt>
          <b>Sl</b>
        </tt>: linear system (<tt>
          <b>syslin</b>
        </tt> list)</li>
      <li>
        <tt>
          <b>n</b>
        </tt>: dimension of unobservable subspace</li>
      <li>
        <tt>
          <b>k</b>
        </tt>: dimension of unstable, unobservable subspace (<tt>
          <b> k&lt;=n</b>
        </tt>).</li>
      <li>
        <tt>
          <b>U</b>
        </tt>: orthogonal matrix</li>
      <li>
        <tt>
          <b>Sld</b>
        </tt>: linear system (<tt>
          <b>syslin</b>
        </tt> list)</li>
      <li>
        <tt>
          <b>tol</b>
        </tt>: threshold for controllability test.</li>
    </ul>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
    Detectability test for <tt>
        <b>sl</b>
      </tt>, a linear system in state-space
    representation. 
    <tt>
        <b>U</b>
      </tt> is a basis whose <tt>
        <b>k</b>
      </tt> first columns span the
    unstable, unobservable subspace of <tt>
        <b>Sl</b>
      </tt> (intersection
    of unobservable subspace of <tt>
        <b>(A,C)</b>
      </tt> and unstable subspace
    of <tt>
        <b>A</b>
      </tt>). Detectability means <tt>
        <b>k=0</b>
      </tt>.</p>
    <p>
      <tt>
        <b>Sld = (U'*A*U,U'*B,C*U,D)</b>
      </tt> displays the "detectable part"
    of <tt>
        <b>Sl=(A,B,C,D)</b>
      </tt>, i.e.</p>
    <pre>

         [*,*,*]
U'*A*U = [0,*,*]
         [0,0,*]
 
C*U = [0,0,*]
   
    </pre>
    <p>
    with <tt>
        <b>(A33,C3)</b>
      </tt> observable (dimension <tt>
        <b>nx-n</b>
      </tt>), <tt>
        <b>A22</b>
      </tt> stable 
    (dimension <tt>
        <b>n-k</b>
      </tt>) and <tt>
        <b>A11</b>
      </tt> unstable (dimension <tt>
        <b>k</b>
      </tt>).</p>
    <h3>
      <font color="blue">Examples</font>
    </h3>
    <pre>

A=[2,1,1;0,-2,1;0,0,3];
C=[0,0,1];
X=rand(3,3);A=inv(X)*A*X;C=C*X;
W=syslin('c',A,[],C);
[k,n,U,W1]=dt_ility(W);
W1("A")
W1("C")
 
  </pre>
    <h3>
      <font color="blue">See Also</font>
    </h3>
    <p>
      <a href="contr.htm">
        <tt>
          <b>contr</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="st_ility.htm">
        <tt>
          <b>st_ility</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="unobs.htm">
        <tt>
          <b>unobs</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="stabil.htm">
        <tt>
          <b>stabil</b>
        </tt>
      </a>,&nbsp;&nbsp;</p>
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